Suppose k possible samples of size n can be selected
from a population of size N.
The standard deviation of the sampling distribution is
the "average" deviation between the k sample
means and the true population mean, μ.
The standard deviation of the sample mean
σx is:

σx =
σ *
sqrt{ ( 1/n ) * ( 1 - n/N ) * [ N / ( N - 1 ) ] }

where σ is the standard deviation of the population,
N is the population size, and
n is the sample size. When the population size is much
larger (at least 10 times larger) than the sample size, the
standard deviation can be approximated by:

σx =
σ / sqrt( n )

When the standard deviation of the population σ is unknown,
the standard deviation of the sampling distribution
cannot be calculated. Under these
circumstances, use the standard error.
The standard error (SE) provides an unbiased estimate of the
standard deviation. It can be calculated from the equation below.

SEx =
s *
sqrt{ ( 1/n ) * ( 1 - n/N ) * [ N / ( N - 1 ) ] }

where s is the standard deviation of the sample,
N is the population size, and
n is the sample size. When the population size is much
larger (at least 10 times larger) than the sample size, the
standard error can be approximated by:

SEx =
s / sqrt( n )

Note: In real-world analyses, the standard deviation of the
population is seldom known. Therefore, the standard error is used
more often than the standard deviation.

Alert

The Advanced Placement Statistics
Examination only covers the "approximate" formulas for the standard
deviation and standard error. However, students are expected to be
aware of the limitations of these formulas; namely, the
approximate formulas should only be used when the population
size is at least 10 times larger than the sample size.

Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic+margin of error. And the uncertainty is denoted
by the confidence level.

In the next section, we work through a problem that shows how to use
this approach to construct a confidence interval to
estimate a population mean.

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Test Your Understanding of This Lesson

Problem 1

Suppose a simple random sample of 150 students is drawn
from a population of 3000
college students. Among sampled students, the average IQ score is
115 with a standard deviation of 10. What is the 99%
confidence interval for the students' IQ score?

Since the above requirements are satisfied, we can use the following
four-step approach to construct a confidence interval.

Identify a sample statistic. Since we are trying to estimate
a population mean, we choose the sample mean
(115) as the sample statistic.

Select a confidence level. In this analysis, the confidence level
is defined for us in the problem. We are working with a 99%
confidence level.

Find the margin of error. Elsewhere on this site, we show
how to compute the margin of error when the sampling
distribution is approximately normal. The key steps are
shown below.

Find standard deviation or standard error. Since we do not
know the standard deviation of the population, we cannot compute the
standard deviation of the sample mean; instead, we compute the standard
error (SE). Because the sample size is much smaller than the
population size, we can use the "approximate" formula for the
standard error.

SE =
s / sqrt( n ) = 10 / sqrt(150) = 10 / 12.25 = 0.82

Find critical value. The critical value is a factor used to
compute the margin of error. Because the standard deviation
of the population is unknown, we express the critical
value as a
t score
rather than a
z score.
To find the critical value, we take these steps.